Role of compliant mechanics and motor control in hopping - from human to robot

Compliant leg function found during bouncy gaits in humans and animals can be considered a role model for designing and controlling bioinspired robots and assistive devices. The human musculoskeletal design and control differ from distal to proximal joints in the leg. The specific mechanical properties of different leg parts could simplify motor control, e.g., by taking advantage of passive body dynamics. This control embodiment is complemented by neural reflex circuitries shaping human motor control. This study investigates the contribution of specific passive and active properties at different leg joint levels in human hopping at different hopping frequencies. We analyze the kinematics and kinetics of human leg joints to design and control a bioinspired hopping robot. In addition, this robot is used as a test rig to validate the identified concepts from human hopping. We found that the more distal the joint, the higher the possibility of benefit from passive compliant leg structures. A passive elastic element nicely describes the ankle joint function. In contrast, a more significant contribution to energy management using an active element (e.g., by feedback control) is predicted for the knee and hip joints. The ankle and knee joints are the key contributors to adjusting hopping frequency. Humans can speed up hopping by increasing ankle stiffness and tuning corresponding knee control parameters. We found that the force-modulated compliance (FMC) as an abstract reflex-based control beside a fixed spring can predict human knee torque-angle patterns at different frequencies. These developed bioinspired models for ankle and knee joints were applied to design and control the EPA-hopper-II robot. The experimental results support our biomechanical findings while indicating potential robot improvements. Based on the proposed model and the robot’s experimental results, passive compliant elements (e.g. tendons) have a larger capacity to contribute to the distal joint function compared to proximal joints. With the use of more compliant elements in the distal joint, a larger contribution to managing energy changes is observed in the upper joints.


Linear versus nonlinear springs
In the pursuit of exploring various passive compliance models, we evaluated both quadratic and cubic nonlinear springs in addition to the linear spring.When analyzing the second-order nonlinear spring, we observed that the residual torques closely resembled those of a linear spring.However, when we introduced the cubic nonlinear spring, the most significant improvements were observed in the ankle joint, where residual torque ratio ranged from 10-14% down to 4-6%.Additionally, for lower frequencies, the knee joint exhibited a maximum 15% reduction in residuals.Nevertheless, the results were on par with the performance achieved using FMC and FMC combined with a spring.

Sensitivity Analysis
We developed a simulation model consisting of four segments (foot, shank, thigh, trunk) and three major leg joints to investigate how variations in joint stiffness affect leg stiffness.The segments' lengths were set to the average values found in human experiments.These joints were equipped with constant stiffness springs (ankle, knee, hip).We proceeded by employing the joint stiffness observed in each joint (averaged across all trials) at each frequency and systematically varied the stiffness of one joint while keeping all other parameters constant.We found the resulting leg stiffness at different configurations that the leg accepts in the stance phase of hopping with that specific frequency.We then calculated the average leg stiffness from the model in this movement range for every new combination of joint stiffness.We normalize joint/leg stiffness by dividing their simulation values by the average joint/leg stiffness found in human hopping at corresponding frequencies.Therefore, point (1,1) in each graph represents the human hopping experiment condition in that frequency.Figure S5 illustrates the leg versus joint sensitivity of the leg stiffness to variation of the joint stiffness from 0.5 to 2 times their identified values in human hopping at each frequency.As can be seen, leg stiffness exhibits a high sensitivity (p-value < 0.05) to alterations in ankle joint stiffness while remaining relatively insensitive (p-value > 0.05) to variations in knee or hip stiffness.This suggests that the observed changes in ankle stiffness among the subjects were the most significant mechanism for adapting leg stiffness, which can be seen in correlation coefficients (Table S1).

Figure S1 .Figure S2 .
Figure S1.Comparison of residual torque ratio at different joints between linear and nonlinear springs.The ratio of residual torque to the joint torque for different hopping frequencies following the implementation of a linear spring, a quadratic spring, and a cubic spring.The standard error for each value is shown as an error bar.

Figure S3 .Figure S4 .
Figure S3.Knee work-loop diagrams.The torque-angle relation for the knee joint in different frequencies for all 18 trials is shown in light grey, while two typical behaviors at each frequency are depicted in blue and red.Negative and positive work-loops are observed, showing the insufficiency of the passive elements to replicate the knee behavior and the necessity of active energy management at the knee joint.

Figure
Figure Sensitivity of the model's leg stiffness to changes in joint stiffness.A 4-segmented simulation model with human hopping experimental data at different frequencies is used.

Table S1 .
Correlation coefficient between leg stiffness and each joint stiffness.The calculated correlation coefficients between hip, knee, and ankle joints with leg stiffness for different frequencies and the corresponding p-values are reported.